3.3 Numerical simulations and experimental results
3.3.1 Reflection geometry
In telecommunication, WDM filters are needed to select and/or manipulate a desired wavelength from a bank of available channels. In reconfigurable communication systems, tunable optical filters play an increasingly important part. Examples include tunable arrayed waveguide gratings (AWG) [10], wavelength tuning based on varying temperature [11] and the application of stress [12]. Tunable filters have also been realized in reflection-geometry VHGs by means of angular tuning[13]. To appreciate and efficiently utilize this filtering configuration, it is important to know how oblique incidence impacts the filtering properties.
3.3.1.1 Experimental setup
The experimental setup is schematically shown in Fig. 3.2; a reflection-geometry VHG, denoted by VHGR, whose grating period Λ is about 532 nm (the corresponding Bragg wavelength at normal incidence is about 1581 nm) is mounted on a rotational stage for precise angular control. The light from a tunable laser source (tuning range 1520 nm ∼1600 nm) is channeled through a fiber collimator (Newport model f-col-9-15) and then used to conduct measurements. The interaction length at normal incidence L is 14 mm. The output laser beam profile from the collimator is Gaussian with a diameter of 0.5 mm, which is the spatial width across its intensity profile where it drops to 1/e2 of the peak value. A beam expander (5×) consisting of two cylindrical lenses (a negative lens with focal length -15 mm and a positive lens whose focal length is 75 mm) can be moved in to widen the incident beam. The angular spread of the (un-)expanded beam is calculated to be (0.15◦)0.03◦inside the glass, corresponding to (0.225◦)0.045◦ in the air.
For each incident angleθ and wavelengthλ of interest, the power of both the transmitted and diffracted beams are monitored, from which the diffraction efficiency can be calculated. A razor blade motion-controlled by a translation stage can be moved across the diffracted beam to measure its intensity profile. The distance between the razor blade and the output face is about 50 mm.
3.3.1.2 Wavelength selectivity
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λ [ nm ]
Transmittance [ dB ]
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λ [ nm ]
Transmittance [ dB ]
(a)W = 0.5 mm (b)W= 2.5 mm
Figure 3.3: Reflection geometry. Wavelength selectivity curves from normal to oblique incidence.
The measured curves in both parts (a) and (b) correspond, from right to left, to incident angles 0◦, 1◦, 2◦, ... 19◦ outside the glass sample (about 0◦ to 13.3◦ inside the glass).
To measure wavelength selectivity curves, the incident angle θ is first set to one of a series of predetermined values, from 0◦ to 13.3◦, and then a wavelength scan is carried out. Fig. 3.3(a) shows the results obtained with the narrow/unexpanded beam BN (beam width W = 0.5 mm);
the transmittance (= 1−η) curves are plotted out in dB. Each valley-like feature corresponds to
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θ
Transmittance
Numerical Prediction
Figure 3.4: Summary of the wavelength selectivity measurements. The increasing transmission of the narrow beam at oblique incidence contrasts strongly with the transmission of the expanded beam, which does not increase much at oblique incidence.
strong coupling of the incident beam to the diffracted beam, and the wavelength of its transmis- sion minimum, defined as the Bragg wavelengthλB, is specified by the incident angle through the relationship
λB= 2nΛ cosθ. (3.5)
The same set of measurements done for the wide/expanded beamBW (beam width 2.5 mm) is summarized in Fig. 3.3(b). As we can see, the almost constant transmittance ofBW at oblique inci- dence contrasts strongly with the increasing trend ofBN, which is predicted by numerical simulations as well. The quantitative comparison is further summarized in Fig. 3.4; the numerical simulations (calculated with an index modulation ∆n= 4.7×10−4) agree very well with the experimental data.
3.3.1.3 Angular selectivity
To measure the angular selectivity curves, the incident angle is first set to one of a series of prede- termined values. With the wavelength tuned to the appropriate Bragg wavelengthλB, an angular scan is then performed. Fig. 3.5(a) and Fig. 3.5(b) show the measured angular selectivities forBN
andBW, respectively. The 0.5 dB angular bandwidths ∆θBW are summarized in Fig. 3.6 along with the numerically simulated results.
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θ
Transmittance [ dB ]
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θ
Transmittance [ dB ]
(a)W = 0.5 mm (b)W= 2.5 mm
Figure 3.5: Reflection geometry. Angular selectivity curves from normal to oblique incidence. The solid curves in both parts (a) and (b) correspond, from left to right, to incident angles 0◦, 1◦, 2◦, ...
19◦ outside the glass sample (about 0◦ to 13.3◦ inside the glass). The dashed curves in both plots are measured for an incident angle of 0.5◦ outside the glass.
Again, the minimum transmittance increases much more rapidly forBN than forBW as we tilt the beam from normal to oblique incidence. An interesting feature common to bothBN andBW is the dramatic decrease of the angular bandwidth ∆θBW at oblique incidence.
Two of the angular selectivity curves, as traced out by dashed lines in Figs. 3.5(a) and 3.5(b), near normal incidence have a funny “twin-valley” (ω) shape. Each of them can be thought of as the “fusion” of two normal, single-dip angular selectivity curves positioned close together. Owing to geometrical degeneracy, the effects caused by a positive θ is equivalent to those caused by a negative one in the reflection geometry; whenever θ is small compared with ∆θBW, such “fusion”
will inevitably occur. This is the reason why we observe the increase of ∆θBW prior to its drastic decrease.
The angular and wavelength selectivity curves are not behaving independently. The relationships between them can be unraveled if we consider the spatial harmonic components of the incident beam. At normal incidence, the VHG’s angular bandwidth ∆θBW is wide and diffracts most spatial harmonics contained within both BN and BW; therefore η approaches 100%. If we slightly tilt the incident beam away from the normal, we effectively increase ∆θBW, and almost all spatial harmonics remain strongly diffracted and the wavelength selectivity curve does not change much.
Around normal incidence where we get theω-shaped angular selectivity, the wavelength selectivity ofBN differs little from that ofBW. However, as we tilt the incident beam past an angular threshold (about 1◦in our case), ∆θBW decreases sharply, and only a smaller portion of the spatial harmonic content ofBN gets diffracted efficiently by the grating. At the same time,BW is not affected as much thanks to its narrower spatial frequency spread. At oblique incidenceBW has a higher diffraction efficiency because a bigger part of the energy ofBN spills out of the ∆θBW angular stop band along
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θ [ in deg ]
∆θ BW [ in deg ]
W = 0.5 mm W = 2.5 mm
Simulation for W = 0.5 mm Simulation for W = 2.5 mm
Figure 3.6: Summary of the angular selectivity measurements. The 0.5 dB angular bandwidth,
∆θBW, is plotted against the angle of incidenceθ. Numerical simulations are seen to agree well with the experimental data.
with the undiffracted spatial components. To put it succinctly, the product of a VHG’s angular selectivity curve and the angular spectrum of the incident beam determines the diffraction efficiency and filter shape of the VHG.
3.3.1.4 Diffracted beam profiles
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x [ in mm ]
Reflected Beam Profile
θ = 0o [ Sim ] θ = 2.69o [Sim ] θ = 5.38o [ Sim ] θ = 8.06o [ Sim ] θ = 10.72o [ Sim ] θ = 13.35o [ Sim ] θ = 0o [ Exp ] θ = 2.69o [ Exp ] θ = 5.38o [ Exp ] θ = 8.06o [ Exp ] θ = 10.72o [ Exp ] θ = 13.35o [ Exp ]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
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x [ in mm ]
Reflected Beam Profile
(a)W = 0.5 mm (b)W= 2.5 mm
Figure 3.7: Reflection geometry. Diffracted beam intensity profiles from normal to oblique incidence.
In Fig. 3.7(a) and Fig. 3.7(b), we show the experimentally measured and numerically simulated diffracted beam intensity profiles forBN andBW, respectively. The legend in Fig. 3.7(a) applies to both plots. At oblique incidence, we see that the diffracted beam ofBN is flattened out from the ideal Gaussian profile. On the contrary, the diffracted beams of BW maintain their Gaussian-like profiles. This phenomenon is attributed to the strong angular filtering suffered byBN thanks to the VHG: its wider diffracted beam profiles are results of less diffracted spatial harmonic components.
This effect is not readily observable for BW because most of its spatial harmonics are diffracted.
Again, the simulations can accurately predict the measured data.